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For a finite monoid S, let ν(S) (νd(S)) denote the least number n such that there exists a graph (directed graph) Γ of order n with End(Γ)≅S. Also let rank(S) be the smallest number of elements required to generate S. In this paper, we use Cayley digraphs of monoids, to connect lower bounds of ν(S) (νd(S)) to the lower bounds of rank(S). On the other hand, we connect upper bounds of rank(S) to upper bounds of ν(S) (νd(S)). [ABSTRACT FROM AUTHOR]

Extending the notions of inverse transversal and associate subgroup, we consider a regular semigroupSwith the property that there exists a subsemigroupTwhich contains, for eachx∈S, a uniqueysuch that bothxyandyxare idempotent. Such a subsemigroup is necessarily a group which we call aspecial subgroup. Here, we investigate regular semigroups with this property. In particular, we determine when the subset of perfect elements is a subsemigroup and describe its structure in naturally arising situations. [ABSTRACT FROM AUTHOR]

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4. Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups. [ABSTRACT FROM AUTHOR]

We study semigroups that behave nicely with respect to a distinguished subset of idempotents E, both in terms of the extended Green's relations $$\widetilde{\mathcal {K}}_E$$ and as unary semigroups. New structure theorems are given, notably in the case of central idempotents. Finally, the decomposition theorems are applied to the study of regular semigroups with particular generalized inverses. [ABSTRACT FROM AUTHOR]

Let, whereSnis the symmetric group onXnandPKn,ris the semigroup of partial mapssuch that |im(α)| ≤ rfor 1 ≤ r ≤ n − 1. In this paper, we find necessary and sufficient conditions for any subset ofPKn,rto be a (minimal) relative generating set of the subsemigroupPTn,rmoduloSn. Then, for each 1 ≤ r ≤ n − 1, we show that the smallest number of elements ofPKn,rwhich, together withSn, generatePTn,riswherepr(n − s) is the number of partitions ofn − swithrterms. We also consider, whereAnis the alternating group onXn. [ABSTRACT FROM AUTHOR]

Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form $$\tau : A^G \rightarrow A^G$$ (where G is any group and A is any set), the theory of cellular automata has been greatly enriched by its connections with group theory and topology. In this paper, we begin the finite semigroup theoretic study of cellular automata by investigating the rank (i.e. the cardinality of a smallest generating set) of the semigroup $$\mathrm {CA}(\mathbb {Z}_n; A)$$ consisting of all cellular automata over the cyclic group $$\mathbb {Z}_n$$ and a finite set A. In particular, we determine this rank when n is equal to p, $$2^k$$ or $$2^kp$$ , for any odd prime p and $$k \ge 1$$ , and we give upper and lower bounds for the general case. [ABSTRACT FROM AUTHOR]

An ordered regular semigroupSisE-specialif for everyx ∈ Sthere is a biggestx+ ∈ Ssuch that bothxx+andx+xare idempotent. Every regular strong Dubreil–Jacotin semigroup isE-special, as is every ordered completely simple semigroup with biggest inverses. In anE-special ordered regular semigroupSin which the unary operationx → x+is antitone the subsetPof perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal ofPif and only ifShas a biggest idempotent. IfS+is a subsemigroup andSdoes not have a biggest idempotent, thenPcontains a copy of the crown bootlace semigroup. [ABSTRACT FROM PUBLISHER]

Let $$S$$ be a semigroup and ' $$*$$ ' a unary operation on $$S$$ which satisfies the following identities Then, $$S^*=\{x^*|x\in S\}$$ is called a regular $$*$$ - transversal of $$S$$ in the literatures. Following Munn and Hall's idea, in this paper we construct fundamental regular semigroups with quasi-ideal regular $$*$$ -transversals by which fundamental representations of regular semigroups with quasi-ideal regular $$*$$ -transversals are obtained. [ABSTRACT FROM AUTHOR]

LetPnandTnbe the partial transformation and the full transformation semigroups on the set {1,…,n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of heightrin the subsemigroupPK(n,r) = {α ∈Pn: |im (α)| ≤r} ofPnto be a (minimal) generating set ofPK(n,r); and similarly, for any set of full transformations of heightrin the subsemigroupK(n,r) = {α ∈Tn: |im (α)| ≤r} ofTnto be a (minimal) generating set ofK(n,r) for 2 ≤ r ≤ n − 1. [ABSTRACT FROM AUTHOR]

A characterization of orthodox transversals is given and the maximum idempotent-separating congruence concerned with quasi-ideal orthodox transversals is explored. The well-known results about orthodox semigroups are generalized and enriched. [ABSTRACT FROM AUTHOR]

In this paper, we will characterize good congruences on abundant semigroups with $$PSQ$$ -adequate transversals by using the so called good congruence triple, which generalizes some results on abundant semigroups with multiplicative adequate transversals. [ABSTRACT FROM AUTHOR]

Generalized orthodox transversals are introduced in this article, and some characterizations associated with them are obtained. The related results of Chen on quasi-ideal orthodox transversals obtained in 1999 and Zhang and Wang on generalized inverse transversals obtained in 2008 are generalized and amplified. A structure theorem of a regular semigroup with a perfect generalized orthodox transversal is also established. [ABSTRACT FROM PUBLISHER]

Let T( X) be the full transformation semigroup on the set X and let T( X, Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. In 2011, Sanwong studied the regular part of T( X, Y) and described its Green's relations and ideals. In this paper, we compute the rank of F( X, Y) when X is a finite set. Moreover, we obtain the rank and idempotent rank of its ideals. [ABSTRACT FROM AUTHOR]

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands. [ABSTRACT FROM PUBLISHER]

In this article, we investigate the ranks of SOP n , SPOP n , and SSPOP n (the semigroups of orientation preserving singular selfmaps, partial, and strictly partial transformations on [n] = {1, 2,…, n}, respectively). Firstly, we show that the rank and idempotent rank of SOP n are n. Secondly, we characterize the structure of the idempotent-generating sets of SPOP n , and prove that the rank and idempotent rank of SPOP n are 2n. Finally, we find that the rank of SSPOP n is n + 1. This research extends the results of Gomes and Howie [4] on the ranks of the semigroups O n , PO n , and SPO n (the semigroups of order preserving singular selfmaps, partial, and strictly partial transformations on [n] = {1, 2,…, n}, respectively). [ABSTRACT FROM AUTHOR]

In this paper, some properties of good homomorphic images of abundant semigroups with a multiplicative quasi-adequate transversal are obtained. After that, good congruences on a class of abundant semigroups with such transversals are studied. We characterize the good congruences by using the so-called good congruence triple. Furthermore, we prove that any good congruence on a class of abundant semigroups with such transversals can be constructed by the good congruence triple. [ABSTRACT FROM AUTHOR]

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x ∈ S, in relation to the natural partial order ≤S in S. In this article, we investigate a class of orthodox monoids with an associate inverse subsemigroup and obtain a known description of uniquely unit regular orthodox semigroups as a corollary. Also, by considering a more general situation, we identify the homomorphic image of a kind of semidirect product of a band with identity by an inverse monoid, thus extending a known result for unit regular orthodox semigroups. [ABSTRACT FROM AUTHOR]

We consider regular semigroups S that are the disjoint union of those local submonoids eSe for which the idempotent e is regularity-preserving. The structure of these is given in terms of Rees matrix semigroups over regular monoids, and a description of certain RP-dominated regular semigroups is obtained as a special case. [ABSTRACT FROM AUTHOR]

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S♭ be multiplicative quasi-adequate transversals of S, and let δ°(δ♭) be the δ-relation on S°(S♭). Then there exists a bijection ϕ from S°/δ°onto S♭/δ♭. In particular, if δ°and δ♭ are congruences, then the bijection ϕ is an isomorphism. [ABSTRACT FROM AUTHOR]

In 1981, Tamura posed a number of problems regarding the axiomatic definition of inverse semigroups. The main goal of this article is to use automated reasoning to solve these problems. In the process, we find some new defining sets of identities for the class of inverse semigroups and provide a single identity for groups in terms of two binary operations. [ABSTRACT FROM AUTHOR]

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x∈ S, in relation to the natural partial order ≤ S. We describe the structure of a regular semigroup with an associate inverse subsemigroup, satisfying two natural conditions. As a particular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element. [ABSTRACT FROM AUTHOR]

We first consider an ordered regular semigroup S in which every element has a biggest inverse and determine necessary and sufficient conditions for the subset S○ of biggest inverses to be an inverse transversal of S. Such an inverse transversal is necessarily weakly multiplicative. We then investigate principally ordered regular semigroups S with the property that S○ is an inverse transversal. In such a semigroup we determine precisely when the set S☆ of biggest pre-inverses is a subsemigroup and show that in this case S☆ is itself an inverse transversal of a subsemigroup of S. The ordered regular semigroup of 2 × 2 boolean matrices provides an informative illustrative example. The structure of S, when S☆ is a group, is also described. [ABSTRACT FROM AUTHOR]

The concept of quasi-adequate transversals is introduced. Some properties and characterizations for abundant semigroups with a multiplicative quasi-adequate transversal are obtained. In particular, a structure theorem for such abundant semigroups is given. Finally, some special cases are considered. [ABSTRACT FROM AUTHOR]

A semigroup S is said to have an associate subgroup G if, for each s ∈ S, there is a unique s* ∈ G such that ss*s = s. If the identity 1G of G is medial, i.e., c1Gc = c holds for each c being a product of idempotents, we show that S is isomorphic to a certain subsemigroup of a semidirect product of an idempotent generated semigroup C by G. If additionally S is orthodox, we may choose C to be a band, belonging to the band variety, generated by the band of idempotents of S. [ABSTRACT FROM AUTHOR]

Let S be a regular semigroup, S o an inverse subsemigroup of S. S o is called a generalized inverse transversal of S, if V ( x) ∩ S o ≠ ø. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases. [ABSTRACT FROM AUTHOR]

A class V of regular semigroups is an e-variety if it is closed under homomorphic images, regular subsemigroups, and direct products. Let S be a regular semigroup and S° an inverse subsemigroup of S. Then S° is called an “inverse transversal of S” if it contains a unique inverse x° of each element x of S. Many important classes of regular semigroups form e-varieties of regular semigroups. However, the class of regular semigroups with inverse transversals does not form an e-variety. In this article, we consider a regular semigroup S with an inverse transversal S° as a regular unary semigroup (S, ○) with a regular unary operation “○” on S firstly. Then we prove that S is a regular semigroup with an inverse transversal S° if and only if (S, ○) satisfies the following identities (IST): [image omitted] Such a regular operation is called an “ist-operation,” and a regular semigroup S is called an “ist-semigroup” if there exists an ist-operation “○” on S. A regular subsemigroup T of a regular semigroup S is called an “ist-subsemigroup” if T is an ist-semigroup. A class V of ist-semigroups is an ist-variety if it is closed under homomorphic images, ist-subsemigroups, and direct products. We characterize the set of identities of (IST) and investigate the relationship among those identities. Also, we describe the classes of regular unary semigroups which satisfy some of these identities in (IST). On the basis of that, we'll characterize the ist-varieties, in a later article. [ABSTRACT FROM AUTHOR]

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S. [ABSTRACT FROM AUTHOR]

The notion of an inverse transversal of a regular semigroup is well-known. Here we investigate naturally ordered regular semigroups that have an inverse transversal. Such semigroups are necessarily locally inverse and the inverse transversal is a quasi-ideal. After considering various general properties that relate the imposed order to the natural order, we highlight the situation in which the inverse transversal is a monoid. The regularity of Green’s relations is also characterised. Finally, we determine the structure of a naturally ordered regular semigroup with an inverse monoid transversal. [ABSTRACT FROM AUTHOR]

In this paper, we investigate idempotent separating and arbitrary homomorphic images of semidirect products of bands by groups. We give characterizations for idempotent separating homomorphic images of semidirect products, and show that the class of all idempotent separating homomorphic images is strictly contained in the class of all homomorphic images. Furthermore, we give a characterization of all homomorphic images. [ABSTRACT FROM AUTHOR]

The aim of this article is to study abundant semigroups with generalized adequate transversals. We obtain some properties of such semigroups and give, in particular, a construction of the class of abundant semigroups with quasi-ideal generalized adequate transversals. Finally, we apply this construction to some special cases. [ABSTRACT FROM AUTHOR]

A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup ℳ 0 [ G ; I , Λ; P ], is given. The result generalizes that of Ruškuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of | I |, |Λ|, the number of connected components k of P , and a number r min , which we define. We go on to show that the number r min is expressible in terms of a family of subgroups of G , the members of which are in one-to-one correspondence with, and determined by the nonzero entries of, the components of P . A number of applications are given, including a generalization of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B ( G ,{1,..., n }). [ABSTRACT FROM AUTHOR]

The investigation of regular F -abundant semigroups is initiated. In fact, F -abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F -abundant semigroups is obtained. In addition, we also prove that a regular F -abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F -inverse semigroups. [ABSTRACT FROM AUTHOR]

A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group of permutations on N elements and by a set of transformations of rank N-1. The aim of this paper is to determine computationally efficient conditions to test whether or not a near permutation semigroup is regular. [ABSTRACT FROM AUTHOR]

Let Tn be the full transformation semigroup on the n-element set Xn. For an arbitrary integer r such that 2 ≤ r ≤ n-1, we completely describe the maximal subsemigroups of the semigroup K(n, r) = {α ∈ Tn : |im α| ≤ r}. We also formulate the cardinal number of such subsemigroups which is an answer to Problem 46 of Tetrad in 1969, concerning the number of subsemigroups of Tn. [ABSTRACT FROM AUTHOR]

The rank of a finite semigroup is the minimal size of its generating set. Let Tn be the semigroup of all total transformations of the set {1, 2,…, n}, and let [Oscr]n be the subsemigroup of Tn of all the order-preserving transformations whose images consist of at most n - 1 elements. We study the subsemigroups S[Oscr](τ) of [Oscr]n generated by the order-preserving transformations whose kernels are partitions of Xn of a given partition type τ. We characterize idempotent-generated semigroups S[Oscr](τ), and show that S[Oscr](τ) is idempotent-generated precisely when it is regular, and that occurs if and only if the weight r of τ is n - 1 or 1. For arbitrary partition type τ of weight r we determine the Green's relations on S[Oscr](τ), and we show that its rank equals to . [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate abundant semigroups with a multiplicative adequate transversal. Some properties and characterizations for such semigroups are obtained. In particular, we establish the structure of this class of abundant semigroups in terms of left normal bands, right normal bands and adequate semigroups with some simple compatibility conditions. Finally, we apply this structure to some special cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study idempotent-connected abundant semigroups which are disjoint unions of quasi-ideal adequate transversals. After obtaining some characterizations of such semigroups, we establish the structure of this class of semigroups. In addition, we also consider several special cases. [ABSTRACT FROM AUTHOR]